Solve {l}{p+q=k}{2p-q=k+1}{q-p=3k+6}

$$\left. \begin{array} { l } { p + q = k } \\ { 2 p - q = k + 1 } \\ { q - p = 3 k + 6 } \end{array} \right.$$

Solve for p, q, k

$$p=-1$$

$$q=-1$$

$$k=-2$$

Short Steps Using Substitution

Solve p+q=k for k.

$$k=p+q$$

Substitute p+q for k in the second and third equation.

$$2p-q=p+q+1 q-p=3\left(p+q\right)+6$$

Solve these equations for q and p respectively.

$$q=-\frac{1}{2}+\frac{1}{2}p p=-\frac{3}{2}-\frac{1}{2}q$$

Substitute -\frac{1}{2}+\frac{1}{2}p for q in the equation p=-\frac{3}{2}-\frac{1}{2}q.

$$p=-\frac{3}{2}-\frac{1}{2}\left(-\frac{1}{2}+\frac{1}{2}p\right)$$

Solve p=-\frac{3}{2}-\frac{1}{2}\left(-\frac{1}{2}+\frac{1}{2}p\right) for p.

$$p=-1$$

Substitute -1 for p in the equation q=-\frac{1}{2}+\frac{1}{2}p.

$$q=-\frac{1}{2}+\frac{1}{2}\left(-1\right)$$

Calculate q from q=-\frac{1}{2}+\frac{1}{2}\left(-1\right).

$$q=-1$$

Substitute -1 for q and -1 for p in the equation k=p+q.

$$k=-1-1$$

Calculate k from k=-1-1.

$$k=-2$$

The system is now solved.

$$p=-1 q=-1 k=-2$$